Statistical Inference for Managers Probability by Imran Khan

Probability • A sample is a selection of just some items from the population. • A random sample is a sample in which each individual item in the population is equally likely to be drawn.

Statistical Inference for Managers Random Experiment: Any experiment which has many possible outcomes and we are not certain that which possible outcome will occur. Sample Space: Is a list/ set of all possible outcomes of the random experiment. Example: Rolling a dice- SS=1, 2, 3, 4, 5, 6 Tossing a coin- SS= H, T Q Two coins are tossed together. Write SS? SS= (HH, HT, TH, TT) Write SS for toss of three coins together? SS= (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT)

Statistical Inference for Managers Two dices are rolled together. Write SS? (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)

Statistical Inference for Managers Probability of an event= No. of outcomes in favor of that event/ Total no. of possible outcomes Q 3 coins are tossed together. Find i)Probability that there is one head? ii)Probability of at least one head? iii)Probability of two heads? Q Two dices are rolled together. i)What is the probability that sum is 9? = 4/36 ii)What is the probability that first dice shows even number and second shows>4? =6/36

Statistical Inference for Managers Example: Two dices are rolled together. Let A=No. on first dice>4 B= No. on second dice is even C= Sum of scores on both dice <5 i) Decide whether A & B are independent or not? ii) Whether A & C are independent or not? iii) Whether A & C are mutually exclusive or not?

Statistical Inference for Managers P(AᴜB)=P(A)+P(B)-P(AᴒB)- at least, either, or =1 -P(AᴜB)- neither, nor Conditional Probability: If the probability of an Event A is asked giving additional information about some other Event B which has already occurred then it’s known as Conditional Probability of A. P(A|B)= P(AᴒB)/ P(B)

Statistical Inference for Managers Example: Three coins are tossed. Find i) Probability that there are two heads? P(H=2)= 3/8 ii) Probability of two heads given first coin has shown a head? P(H=2|first coin head)= 1/2

Statistical Inference for Managers If X= -3, -2, -1, 0, 1, 2, 3 Prob =0. 05, 0. 1, 0. 3, 0. 1, 0. 2 i) P(X<2)= ? ii) P(X²>4)= ? iii) P(X is even|X<2)= ?